1.15 Phase replacement and dephasing are different operations
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The unsigned endpoint is a coherent state with specified positive amplitudes. It should not be confused with a state obtained by forgetting all relative phases. The latter is represented by a mixed state, not generally by one state vector.
For a normalized vector \(v\), its density matrix is \(vv^\dagger\). Complete dephasing in the sample basis keeps only its diagonal entries. In the present example, dephasing either \(\psi\) or \(\chi\) gives the same diagonal matrix
Its trace is one, and every diagonal entry is nonnegative.
Theorem QA20: The dephased selected probability
The selected projective-measurement probability of this diagonal state is
Proof
Every diagonal entry of \(P\) is \(c(0)/12=4/12=1/3\). Multiplication by a diagonal matrix and taking the trace gives
\(\square\)
The three outcomes are therefore distinct: the signed coherent state has selected probability one, the unsigned coherent state has probability zero, and the completely dephased state has probability one third. This calculation prevents an incorrect identification of amplitude sign removal with physical decoherence. QA20 is a general finite-dimensional calculation specialized to the example and proved here. Density matrices and dephasing are outside the inherited Lean package.